Dilogarithm Identities
Abstract
We study the dilogarithm identities from algebraic, analytic, asymptotic, K-theoretic, combinatorial and representation-theoretic points of view. We prove that a lot of dilogarithm identities (hypothetically all!) can be obtained by using the five-term relation only. Among those the Coxeter, Lewin, Loxton and Browkin ones are contained. Accessibility of Lewin's one variable and Ray's multivariable (here from n ≤ 2 only) functional equations is given. For odd levels and hat{sl_2} case of Kuniba-Nakanishi's dilogarithm conjecture is proven and additional results about remainder them are obtained. The connections between dilogarithm identities and Rogers-Ramanujan-Andrews-Gordon type partition identities via their asymptotic behavior are discussed. Some new results about the string functions for level k vacuum representation of the affine Lie algebra hat{sl_n} are obtained. Connection between dilogarithm identities and algebraic K-theory (torsion in K_3(R)) is discussed. Relations between crystal bases, branching functions b_λ^{kΛ_0}(q) and Kostka-Foulkes polynomials (Lusztig's q-analog of weight multiplicity) are considered. The Melzer and Milne conjectures are proven. In some special cases we are proving that the branching functions b_λ^{kΛ_0}(q) are equal to an appropriate limit of Kostka polynomials (the so-called Thermodynamic Bethe Ansatz limit). Connection between ``finite-dimensional part of crystal base'' and Robinson-Schensted-Knuth correspondence is considered.
- Publication:
-
Progress of Theoretical Physics Supplement
- Pub Date:
- 1995
- DOI:
- 10.1143/PTPS.118.61
- arXiv:
- arXiv:hep-th/9408113
- Bibcode:
- 1995PThPS.118...61K
- Keywords:
-
- High Energy Physics - Theory;
- Mathematics - Classical Analysis and ODEs;
- Mathematics - Quantum Algebra
- E-Print:
- 96 pages